5

This a computational challenge, to find an efficient algorithm to discover a quadruple $(n,n+1,n+2,n+3)$ with the same sum of prime factors as described in the MO question, "Ruth-Aaron triples, etc." E.g., $$417,164 = 2^2 \cdot 11 \cdot 19 \cdot 499 \;;\; 2+2+11+19+499 = 533 \;.$$ The sum can be computed by multiplying the exponent in the prime factorization times the base prime:

```
SumFact[n_] := Apply[Plus, Map[#[[1]] #[[2]] &, FactorInteger[n]]];
```

Apparently no such quadruple is known, and I've checked through $n=10^7$, and am now trying to reach $10^8$ in the next day or so. But my computation is naive in terms of efficient computation. Also, I do not have easy access to significant computational resources.

As far as I can make out, there is no known such quadruple.

1Oh, there is an indication that such a quadruple must lie beyond $10^{10}$, which is well beyond what resources I can muster. – Joseph O'Rourke – 2014-12-24T03:03:20.243

2You need only search first every successive pairs of even numbers n to find adjacent even Ruth-Aaron numbers; if found, only then test the three additional adjacent (odd) numbers. This eliminates roughly half the computations in the naive method. Next, because in this first sieve search you're only searching even numbers, you could instead divide each by 2 (which subtracts the same increment in the R-A sum), and so instead search successive pair of "half" numbers. Finally, this search is highly parallelizable using Parallelize[]. – David G. Stork – 2014-12-24T17:58:27.120